Applying Ito's formula to a complicated expression

Question

I am faced with some (predictable) process $(r_t)$ and let $0 \leq t \leq T$. I am baffled with the issue of applying Ito's formula to the process $$ \bigg\{ \int_{t}^{T} G(s-t, r_t) \,ds \bigg\}_{t \in [0,T]},$$ where $G:[0,T] \times \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function. Any suggestions?

Answer 1

Seeing as your process is made up of Riemann integrals and the function $G$ is smooth, I would try differentiating under the integral sign to get the relevant partial derivatives, which can then be plugged into Ito's lemma. So, for instance, if we label the process $X_t$, then

$$ \begin{eqnarray*} \dfrac{\partial}{\partial t}X_t&{}={}&\dfrac{\partial}{\partial t}\left(\,\,\int\limits_{t}^{T}G(s-t, r_s)\ \mathrm{d}s\right)\newline &&\newline &{}={}&\int\limits_{t}^{T}\left(\dfrac{\partial}{\partial t}G\right)\mathrm{d}s{}-{}G(0,r_t)\,,\newline \end{eqnarray*} $$ ... and so on.